This paper deals with the problem of estimating the regions of attraction (or Safe regions) of two well-known nonlinear tumor growth models by redefining each model as a set of independent artificial systems and utilizing the novel concept of individual invariance. The first model describes leukemia growth in the presence of chemotherapy where the stable tumor-free equilibrium point vanishes once the treatment is stopped, therefore, we define a safe region around the tumor-free equilibrium point. Then, we compute its region of attraction corresponding one treatment protocol. The second model, known as the Stepanova model, describes immune tumor interactions with chemo- and immunotherapy. First, an initial estimate of an invariant set around its tumor-free equilibrium without therapeutic effects is provided. The estimate is used in a second step to validate the invariant set for a higher-order model with chemo- and immunotherapy. Moreover, a parametric sensitivity analysis is carried out to study the effect of model parameter uncertainties on the size of the region of attraction (RoA) for both models. The proposed model restructuring and the associated analysis method prove to be accurate, scalable, and efficient. It allows for sensitivity analysis and an analytical description of the safe regions in tumor growth models giving an insightful look at the tumor dynamics in association with the treatment protocols.

中文翻译：

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化疗和免疫治疗对肿瘤模型稳定区的影响

本文通过将每个模型重新定义为一组独立的人工系统并利用个体不变性的新概念，解决了估计两个著名的非线性肿瘤生长模型的吸引区域（或安全区域）的问题。第一个模型描述了化疗条件下白血病的生长，一旦停止治疗，稳定的无肿瘤平衡点就会消失，因此，我们在无肿瘤平衡点周围定义了一个安全区域。然后，我们计算其对应的一种治疗方案的吸引区域。第二种模型称为 Stepanova 模型，描述免疫肿瘤与化疗和免疫治疗的相互作用。首先，提供了围绕无治疗效果的无肿瘤平衡的不变集的初始估计。该估计值用于第二步，以验证化疗和免疫疗法高阶模型的不变集。此外，还进行了参数敏感性分析，以研究模型参数不确定性对两个模型的吸引区域 (RoA) 大小的影响。所提出的模型重组和相关的分析方法被证明是准确的、可扩展的和高效的。它允许对肿瘤生长模型中的安全区域进行敏感性分析和分析描述，从而深入了解与治疗方案相关的肿瘤动态。